Blender VSE_can I make movie strip gradually slow down or gradually speed up?
Yes, add a speed effect to the selected movie strip. Turn off the option "Stretch to input strip length", then keyframe the value Speed Factor.


Start with a value of 1.0.
Then move to the next frame that should be normal speed and keyframe
that 1.0 as well.
Go to the frame that should be stretched out, change the Speed
Factor value to a smaller value for slow e.g. 0.5 is half speed, or
a higher value e.g. 2.0 is double speed.
You won't see any effect until you press the Refresh Sequencer button at the bottom of the VSE timeline.




Alternatively if you don't need to ramp the clip's speed you could apply a "Stretch to Length" effect. 


Place your play head at the point you want to start slowing down.
Select the Strip and make a hard cut (Shift+K key)
Make another hard cut (Shift+K key) at the end of the region of slow
motion. Select the middle strip and add a Speed Effect, leave it in
"Stretch to input Length" mode.
Move the cut strip up to the channel above and stretch it to the
right length, the speed effect will slow the clip down.
Select the third piece of the clip and move it to the end of the
speed effect, this will continue to play at the original speed.